Introduction to Partial Differential Equations

with MATLAB

Jeffery Cooper

1998 Birkhäuser

Boston • Basel • Berlin

Contents

Preface xiii

1 Preliminaries 1 1.1 Elements of analysis 1

1.1.1 Sets and their boundaries 1 1.1.2 Integration and differentiation 3 1.1.3 Sequences and series of functions 5 1.1.4 Functions of several variables 11

1.2 Vector Spaces and linear Operators 14 1.3 Review of facts about ordinary differential equations 17

2 First-Order Equations 19 2.1 Generalities 19 2.2 First-order linear PDE's 21

2.2.1 Constant coefficients 22 2.2.2 Spatially dependent velocity of propagation 25

2.3 Nonlinear conservation laws 30 2.4 Linearization 39 2.5 Weak Solutions 41

2.5.1 The notion of a weak Solution 41 2.5.2 Weak Solutions of ut + F(u)x -0 43 2.5.3 The Riemann problem 45 2.5.4 Formation of shock waves 47 2.5.5 Nonuniqueness and stability of weak Solutions 48

2.6 Numerical methods 53 2.6.1 Difference quotients 53 2.6.2 A finite difference scheme 55 2.6.3 An upwind scheme and the CFL condition 57 2.6.4 A scheme for the nonlinear conservation law 60

2.7 A conservation law for cell dynamics 64 2.7.1 A nonreproducing model 64

Vlll Contents

2.7.2 The mitosis boundary condition 67 2.8 Projects 70

3 Diffusion 73 3.1 The diffusion equation 73 3.2 The maximum principle 77 3.3 The heat equation without boundaries 81

3.3.1 The fundamental Solution 81 3.3.2 Solution of the initial-value problem 85 3.3.3 Sources and the principle of Duhamel 89

3.4 Boundary value problems on the half-line 95 3.5 Diffusion and nonlinear wave motion 101 3.6 Numerical methods for the heat equation 105 3.7 Projects 110

4 Boundary Value Problems for the Heat Equation 111 4.1 Separation of variables 111 4.2 Convergence of the eigenfunction expansions 116 4.3 Symmetrie boundary conditions 130 4.4 Inhomogeneous problems and asymptotic behavior 141 4.5 Projects 153

5 Waves Again 157 5.1 Acoustics 157

5.1.1 The equations of gas dynamics 157 5.1.2 The linearized equations 159

5.2 The vibrating string 160 5.2.1 The nonlinear model 160 5.2.2 The linearized equation 163

5.3 The wave equation without boundaries 165 5.3.1 The initial-value problem and d'Alembert's formula . . . 165 5.3.2 Domains of influence and dependence 170 5.3.3 Conservation of energy on the line 170 5.3.4 An inhomogeneous problem 174

5.4 Boundary value problems on the half-line 181 5.4.1 d'Alembert's formula extended 181 5.4.2 A transmission problem 186 5.4.3 Inhomogeneous problems 187

5.5 Boundary value problems on a finite interval 192 5.5.1 A geometric construetion 192 5.5.2 Modes of Vibration 193 5.5.3 Conservation of energy for the finite interval 196 5.5.4 Other boundary conditions 198 5.5.5 Inhomogeneous equations 199

Contents ix

5.5.6 Boundary forcing and resonance 201 5.6 Numerical methods 208 5.7 A nonlinear wave equation 211 5.8 Projects 217

6 Fourier Series and Fourier Transform 219 6.1 Fourier series 219 6.2 Convergence of Fourier series 223 6.3 The Fourier transform 231 6.4 The heat equation again 236 6.5 The discrete Fourier transform 238

6.5.1 The DFT and Fourier series 238 6.5.2 The DFT and the Fourier transform 244

6.6 The fast Fourier transform (FFT) 250 6.7 Projects 257

7 Dispersive Waves and the Schrödinger Equation 259 7.1 Oscillatory integrals and die method of stationary phase 259 7.2 Dispersive equations 263

7.2.1 The wave equation 263 7.2.2 Dispersion relations 264 7.2.3 Group velocity and phase velocity 267

7.3 Quantum mechanics and die uncertainty principle 274 7.4 The Schrödinger equation 278

7.4.1 The dispersion relation of the Schrödinger equation . . . 278 7.4.2 The correspondence principle 281 7.4.3 The initial-value problem for me free Schrödinger equation 282

7.5 The spectrum of die Schrödinger Operator 287 7.5.1 Continuous spectrum 287 7.5.2 Bound states of the Square well potential 290

7.6 Projects 296

8 The Heat and Wave Equations in Higher Dimensions 297 8.1 Diffusion in higher dimensions 297

8.1.1 Derivation of die heat equation 297 8.1.2 The fundamental Solution of die heat equation 298

8.2 Boundary value problems for the heat equation 304 8.3 Eigenfunctions for the rectangle . 310 8.4 Eigenfunctions for die disk 315 8.5 Asymptotics and steady-state Solutions 322

8.5.1 Approach to die steady State 322 8.5.2 Compatibility of source and boundary flux 325

8.6 The wave equation 331 8.6.1 The initial-value problem 331

X Contents

8.6.2 Themethodofdescent 335 8.7 Energy 339 8.8 Sources 343 8.9 Boundary value problems for the wave equation 347

8.9.1 Eigenfunction expansions 347 8.9.2 Nodal curves 349 8.9.3 Conservation of energy 349 8.9.4 Inhomogeneous problems 351

8.10 The Maxwell equations 356 8.10.1 The electric and magnetic fields 356 8.10.2 The initial-value problem 359 8.10.3 Plane waves 359 8.10.4 Electrostatics 360 8.10.5 Conservation of energy 361

8.11 Projects 365

9 Equilibrium 367 9.1 Harmonie funetions 367

9.1.1 Examples 367 9.1.2 The mean value property 368 9.1.3 The maximum principle 372

9.2 The Dirichlet problem 377 9.2.1 Fourier series Solution in the disk 377 9.2.2 Liouville's theorem 384

9.3 The Dirichlet problem in a rectangle 389 9.4 The Poisson equation 394

9.4.1 The Poisson equation without boundaries 394 9.4.2 The Green's funetion 399

9.5 Variational methods and weak Solutions 414 9.5.1 Problems in variational form 414 9.5.2 The Rayleigh-Ritz procedure 417

9.6 Projects 423

10 Numencal Methods for Higher Dimensions 425 10.1 Finite differences 425 10.2 Finite elements 433 10.3 Galerkin methods 442 10.4 A reaction-diffusion equation 450

11 Epilogue: Classification 455

Contents XI

Appendices

A Recipes and Formulas 459 A.l Separation of variables in space-time problems 459 A.2 Separation of variables in steady-state problems 464 A.3 Fundamental Solutions 471 A.4 The Laplace Operator in polar and spherical coordinates 474

B Elements of MATLAB 477 B.l Forming vectors and matrices 477 B.2 Operations on matrices 480 B.3 Array Operations 481 B.4 Solution of linear Systems 482 B.5 MATLAB functions and mfiles 483 B.6 Script mfiles and programs 485 B.7 Vectorizing computations 486 B.8 Function functions 488 B.9 Plotting 2-D graphs 490 B.10 Plotting 3-D graphs 492 B.ll Movies 496

C References 497

D Solutions to Selected Problems 501

E List of Computer Programs 527

Index 533